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Print version ISSN 0041-6932

Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca June 2009

Exponents of Modular Reductions of Families of Elliptic Curves

Igor E. Shparlinski

Abstract. For some natural families of elliptic curves we show that "on average" the exponent of the point group of their reductions modulo a prime grows as .

2000 Mathematics Subject Classification. 11B57, 11G07, 14H52

Key words and phrases. Elliptic curves; Group exponent; Farey fractions.

1. Introduction

For integers and such that , we denote by the elliptic curve defined by the affine Weierstraß equation:

For a basic background on elliptic curves, we refer to [11].

For a prime , we denote by the finite field of elements, which we identify with the set of integers .

When , the set , consisting of the -rational points of together with a point at infinity , forms an abelian group under an appropriate composition rule called addition, and the number of elements in the group satisfies the Hasse bound:

 (1)

(see, for example, [11, Chapter V, Theorem 1.1]).

It is well-known that is of rank at most two, that is, is isomorphic to

 (2)

for unique integers and with and . The number is called the exponent of which we denote by . In other words, is the smallest positive such that for all points .

We also put if .

Thus we see that (1) and (2) imply the following trivial bound

 (3)

The exponent of elliptic curves has been studied in a number of works, see [478910], with a variety of results, each of them indicating that in a "typical case" the exponent tends to be substantially larger than the bound (3) (and its analogue for curves over arbitrary finite fields) guarantees.

W. Duke [4], among other results, has proved that, assuming the Generalised Riemann Hypothesis, for every fixed integer and with , and arbitrary small , the bound

 (4)

holds for all but of primes .

It is also shown in [10] that (4 ) holds for all but pairs .

Here we use a combination of the results and ideas of [110] to prove unconditionally that (4 ) is satisfied for almost all pairs with , for and relatively small compared to .

Theorem 1. For any fixed and all integers , satisfying the inequalities

or

the bound

holds for all but pairs with ,

In particular, Theorem 1 is nontrivial if

or

We also show that averaging over gives some additional saving.

Theorem 2. For any fixed and all integers , and satisfying the inequalities

the bound (4 ) holds for all but triples with , , .

We note that the condition from [1], where it is used to simplify the error term, is not neccessary. One can easily extend Theorem 2 for and beyond this range, however since (as in [1]) small values of and are of main interest we have not done this.

We remark that in [5] some of the results of [4] have been extended to hyperelliptic curves. It would also be interesting to obtain analogues of our result for natural families of hyperelliptic curves.

We also consider the set of Farey fractions

In particular

For with and two polynomial , the reduction is correctly defined. Various questions concerning the behaviour of the curves on average over and have been studied in [2]. Here we continue to study this family of curves. Certainly the most interesting case is when is small compared to .

Theorem 3. Assume that the discriminant

is nonzero and the -invariant

is nonconstant. Then for any fixed and all integers and with

the bound

holds for all but pairs with , .

2. Preparations

The following result follows immediately from the more precise statement of [10, Theorem 3.1].

Lemma 4. For any , the number of triples with

is at most .

Let and put

where the maximum is taken over all non-principal multiplicative characters modulo  such that is the principal character .

Similarly, we define and put

where the maximum is taken over all non-principal multiplicative characters modulo  such that is the principal character . For an arbitrary subset , we denote by the number of pairs such that with and . We also denote

The following estimate is given in [1].

Lemma 5. For all primes , integers , and subsets such that whenever the isomorphism implies , the following bound holds:

Moreover, it is shown in [1] that is small "on average" over .

Lemma 6. The following bound holds:

For a prime and an integer with we denote by the number of fractions with and .

It is shown in [3] that is close to its expected value on average over . More precisely, we have:

Lemma 7. We have,

3. Proof of Theorem 1

Let be the set of pairs for which . Then it is enough to show that

Since by Lemma 4 we have , invoking Lemma 5 we see that it is enough to check that .

Assume that then by the Burgess bound, see [6, Theorems 12.5 and 12.6], we have . Also, if then have .

Similarly, if then , and if then have .

4. Proof of Theorem 2

As before, let be the set of pairs for which . Then it is enough to show that

 (5)

Let us assume that since the case is similar.

Using the trivial bound for primes , we deduce

 (6)

Noticing that for the set satisfies the conditions of Lemma 5 , we obtain

 (7)

By Lemma 4 we have

 (8)

Substituting (7) and (8) in (6), we obtain

We now easily verify that under the conditions of the theorem, Lemma 6 implies the desired bound (5).

5. Proof of Theorem 3

As before, we use to denote the set of pairs for which .

Let be the set of such that

for some .

Obviously, for any and we have

(since the corresponding curves are isomorphic, see [11, Section III.1]).

We also note that the system of equations

leads to the equation

which has solutions (by the condition on the -invariant ).

Therefore

Using Lemma 7, we obtain

which concludes the proof.

References

[1]    W. D. Banks and I. E. Shparlinski, 'Sato-Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height', Israel J. Math., (to appear).        [ Links ]

[2]    A. Cojocaru and C. Hall, 'Uniform results for Serre's theorem for elliptic curves', Internat. Math. Res. Notices, 2005 (2005), 3065-3080.        [ Links ]

[3]    A. Cojocaru and I. E. Shparlinski, 'Distribution of Farey fractions in residue classes and Lang-Trotter conjectures on average', Proc. Amer. Math. Soc., 136 (2008), 1977-1986.        [ Links ]

[4]    W. Duke, 'Almost all reductions modulo p of an elliptic curve have a large exponent', Comptes Rendus Mathematique, 337 (2003), 689-692.        [ Links ]

[5]    K. Ford and I. E. Shparlinski, 'On finite fields with Jacobians of small exponent', Preprint, 2006 (available from http://arxiv.org/abs/math.NT/0607474).        [ Links ]

[6]    H. Iwaniec and E. Kowalski, On curves over finite fields with Jacobians of small exponent. Intern. J. Number Theory, 4, 2008, 819-826.        [ Links ]

[7]    F. Luca, J. McKee and I. E. Shparlinski, 'Small exponent point groups on elliptic curves', J. Théorie des Nombres Bordeaux, 18 (2006), 471-476.        [ Links ]

[8]    F. Luca and I. E. Shparlinski, 'On the exponent of the group of points on elliptic curves in extension fields', Intern. Math. Research Notices, 2005 (2005), 1391-1409.        [ Links ]

[9]    R. Schoof, 'The exponents of the group of points on the reduction of an elliptic curve', Arithmetic Algebraic Geometry, Progr. Math., vol. 89, Birkhäuser, Boston, MA, 1991, 325-335.        [ Links ]

[10]    I. E. Shparlinski, 'Orders of points on elliptic curves', Affine Algebraic Geometry, Contemp. Math., vol. 369, Amer. Math. Soc., Providence, RI, 2005, 245-252.        [ Links ]

[11]    J. H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, Berlin, 1995.         [ Links ]

Igor E. Shparlinski
Department of Computing, Macquarie University, North Ryde,
Sydney, NSW 2109, Australia
igor@ics.mq.edu.au

Recibido: 7 de octubre de 2007
Aceptado: 21 de mayo de 2008